Over $\mathbb{C}(T)$ the Isogeny theorem doesn't hold, even if you throw away isotrivial components.

In his paper "Arakelov's theorem for Abelian Varieties" Faltings proves the following analogue of Shafarevich:

Let $B$ be complete smooth complex curve and $S\subset B$ a finite subset of points, and $f:X\rightarrow B\backslash{S}$ a family of Abelian varieties. Let $G$ be the fundamental group of $B\backslash{S}$. Fix a point $p\in B\backslash S$ so that we get an action of $G$ on
$H_1(X_p,\mathbb{Z})$ by monodromy.
Now, say $X$ satisfies (A) if
$$End(X)=End_{G}(H_1(X_p,\mathbb{Z})).$$ Note that the LHS also injects into the RHS.

Faltings proves there are finitely many isomorphisms classes $X$ satisfying (A) of a fixed dimension, and he also gives an example of an 8-dimensional family $X$ which doesn't satisfy (A). Now we'll use this $X$ to give a negative answer to the Isogeny theorem:

By a standard argument (see Milne's Abelian Varieties online notes, Chapter IV, Theorem 2.5) if both $X$ and $X\times X$ have at most finitely many isogenous abelian varieties up to isomorphism, by an isogeny of degree a power of some prime $l$, then
$$End(X)\otimes\mathbb{Q}_l=End_G(H_1(X_p,\mathbb{Z})\otimes\mathbb{Q}_l)$$
where the second endomorphism ring is as $\mathbb{Q}_l$ vector spaces. However, the latter is false for our $X$ by assumption, since taking group invariants commutes with base change over fields, hence either $X$ or $X\times X$ don't satisfy the isogeny theorem.

$\textbf{However},$ if $X$ is a family of abelian varieties over the generic point of $B$ which extends to a family away from $S$ (in other words, it only has bad reduction at points of $S$) then every abelian varietiy isogenous to $X$ shares this property, since good reduction is an isogeny invariant. Hence the isogeny therem is true for all $X$ satisfying (A), and this includes all $X$ of dimension at most 3 by a theorem of Deligne
(see Hodge II, 4.4.13), as long as you insist $X$ has no isotrivial subvarieties.

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